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Remarks on the Cauchy functional equation and variations of it

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Abstract

This paper examines various aspects related to the Cauchy functional equation \(f(x+y)=f(x)+f(y)\), a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.

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  1. Department of Mathematics, The Technion - Israel Institute of Technology, 3200003, Haifa, Israel

    Daniel Reem

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Part of this work was done while the author was at the National Institute of Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil, and at the Institute of Mathematical and Computer Sciences (ICMC), University of São Paulo, São Carlos, Brazil.

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Reem, D. Remarks on the Cauchy functional equation and variations of it. Aequat. Math. 91, 237–264 (2017). https://doi.org/10.1007/s00010-016-0463-6

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